Detection of Edges in Spectral Data

Detection of Edges from Nonuniform Fourier Data

“…However, while this poor response to jump discontinuities is bad for reconstruction purposes, it is evidence that the Fourier coefficients of a piecewise-analytic function contain information that can be used for edge detection. Specifically, the presence of the artificial oscillations at the jump discontinuities of f indicate where, in fact, the discontinuities are located, [9,15,22] …”

“…[9,10,15,23] and references therein). In particular, the concentration factor edge detection method, first introduced in [15], accurately finds the jump locations and their associated values by constructing a modified partial Fourier sum.…”

“…[9,10,15,23] and references therein). In particular, the concentration factor edge detection method, first introduced in [15], accurately finds the jump locations and their associated values by constructing a modified partial Fourier sum. In short, the partial Fourier sum of a piecewise-analytic function is modified so that the energy of the partial sum approximation is "concentrated" at the singular support of the underlying function, thereby highlighting its jump discontinuities.…”

“…Instead of interpolating to uniform Fourier coefficients, we extend the concentration factor edge detection method, [15], to work directly on the given nonuniform Fourier data. Hence no interpolation error is incurred, and the numerical algorithm remains well-conditioned.…”

“…This paper is organized as follows. In Section 2 we review the concentration factor edge detection method for approximating the jump function corresponding to a piecewise-smooth function from given uniform Fourier coefficients, [15,16]. In Section 3 we introduce Fourier frames as a way of investigating nonuniform Fourier data.…”